Let us denote the points as follows:
⇒ O = (0, 0, 0)
⇒ A = (2, 1, 1)
⇒ B = (3, 5, –1)
⇒ C = (4, 3, –1)
If two lines of direction ratios (a1, b1, c1) and (a2, b2, c2) are said to be perpendicular to each other. Then the following condition is need to be satisfied:
⇒ a1 . a2 + b1 . b2 + c1 . c2 = 0 ……(1)
Let us assume the direction ratios for line OA be (r1, r2, r3) and BC be (r4, r5, r6)
We know that direction ratios for a line passing through points (x1, y1, z1) and (x2, y2, z2) is (x2 – x1, y2 – y1, z2 – z1).
Let’s find the direction ratios for the line OA
⇒ (r1, r2, r3) = (2 – 0, 1 – 0, 1 – 0)
⇒ (r1, r2, r3) = (2, 1, 1)
Let’s find the direction ratios for the line BC
⇒ (r4, r5, r6) = (4 – 3, 3 – 5, – 1 – (– 1))
⇒ (r4, r5, r6) = (4 – 3, 3 – 5, – 1 + 1)
⇒ (r4, r5, r6) = (1, –2, 0)
Let us check whether the lines are perpendicular or not using (1)
⇒ r1 . r4 + r2 . r5 + r3 . r6 = (2 × 1) + (1 × –2) +(1 × 0)
⇒ r1 . r4 + r2 . r5 + r3 . r6 = 2 – 2 + 0
⇒ r1 . r4 + r2 . r5 + r3 . r6 = 0
Since the condition is clearly satisfied, we can say that the given lines are perpendicular to each other.